File: HapTutorial.aux

package info (click to toggle)
gap-hap 1.70%2Bds-1
  • links: PTS
  • area: main
  • in suites: forky, sid
  • size: 56,612 kB
  • sloc: xml: 16,139; sh: 216; javascript: 155; makefile: 126; ansic: 47; perl: 36
file content (468 lines) | stat: -rw-r--r-- 46,750 bytes parent folder | download
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\@writefile{toc}{\contentsline {chapter}{\numberline {1}\textcolor {Chapter }{Simplicial complexes \& CW complexes}}{7}{chapter.1}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {1.1}\textcolor {Chapter }{The Klein bottle as a simplicial complex}}{7}{section.1.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {1.2}\textcolor {Chapter }{Other simplicial surfaces}}{8}{section.1.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {1.3}\textcolor {Chapter }{The Quillen complex}}{8}{section.1.3}\protected@file@percent }
\citation{ksontini}
\@writefile{toc}{\contentsline {section}{\numberline {1.4}\textcolor {Chapter }{The Quillen complex as a reduced CW\texttt  {\symbol  {45}}complex}}{9}{section.1.4}\protected@file@percent }
\@writefile{brf}{\backcite{ksontini}{{9}{1.4}{section.1.4}}}
\@writefile{toc}{\contentsline {section}{\numberline {1.5}\textcolor {Chapter }{Simple homotopy equivalences}}{9}{section.1.5}\protected@file@percent }
\citation{spreerkhuenel}
\@writefile{toc}{\contentsline {section}{\numberline {1.6}\textcolor {Chapter }{Cellular simplifications preserving homeomorphism type}}{10}{section.1.6}\protected@file@percent }
\@writefile{brf}{\backcite{spreerkhuenel}{{10}{1.6}{section.1.6}}}
\@writefile{toc}{\contentsline {section}{\numberline {1.7}\textcolor {Chapter }{Constructing a CW\texttt  {\symbol  {45}}structure on a knot complement}}{10}{section.1.7}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {1.8}\textcolor {Chapter }{Constructing a regular CW\texttt  {\symbol  {45}}complex by attaching cells}}{11}{section.1.8}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {1.9}\textcolor {Chapter }{Constructing a regular CW\texttt  {\symbol  {45}}complex from its face lattice}}{12}{section.1.9}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {1.10}\textcolor {Chapter }{Cup products}}{13}{section.1.10}\protected@file@percent }
\citation{spreerkhuenel}
\@writefile{brf}{\backcite{spreerkhuenel}{{15}{1.10}{section.1.10}}}
\citation{milnor}
\@writefile{toc}{\contentsline {section}{\numberline {1.11}\textcolor {Chapter }{Intersection forms of $4$\texttt  {\symbol  {45}}manifolds}}{18}{section.1.11}\protected@file@percent }
\@writefile{brf}{\backcite{milnor}{{18}{1.11}{section.1.11}}}
\citation{goncalves}
\@writefile{toc}{\contentsline {section}{\numberline {1.12}\textcolor {Chapter }{Cohomology Rings}}{19}{section.1.12}\protected@file@percent }
\@writefile{brf}{\backcite{goncalves}{{19}{1.12}{section.1.12}}}
\@writefile{toc}{\contentsline {section}{\numberline {1.13}\textcolor {Chapter }{Bockstein homomorphism}}{20}{section.1.13}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {1.14}\textcolor {Chapter }{Diagonal maps on associahedra and other polytopes}}{21}{section.1.14}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {1.15}\textcolor {Chapter }{CW maps and induced homomorphisms}}{21}{section.1.15}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {1.16}\textcolor {Chapter }{Constructing a simplicial complex from a regular CW\texttt  {\symbol  {45}}complex}}{22}{section.1.16}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {1.17}\textcolor {Chapter }{Some limitations to representing spaces as regular CW complexes}}{23}{section.1.17}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {1.18}\textcolor {Chapter }{Equivariant CW complexes}}{24}{section.1.18}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {1.19}\textcolor {Chapter }{Orbifolds and classifying spaces}}{26}{section.1.19}\protected@file@percent }
\newlabel{secOrbifolds}{{1.19}{26}{\textcolor {Chapter }{Orbifolds and classifying spaces}}{section.1.19}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {2}\textcolor {Chapter }{Cubical complexes \& permutahedral complexes}}{31}{chapter.2}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {2.1}\textcolor {Chapter }{Cubical complexes}}{31}{section.2.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {2.2}\textcolor {Chapter }{Permutahedral complexes}}{32}{section.2.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {2.3}\textcolor {Chapter }{Constructing pure cubical and permutahedral complexes}}{34}{section.2.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {2.4}\textcolor {Chapter }{Computations in dynamical systems}}{35}{section.2.4}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {3}\textcolor {Chapter }{Covering spaces}}{36}{chapter.3}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {3.1}\textcolor {Chapter }{Cellular chains on the universal cover}}{36}{section.3.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {3.2}\textcolor {Chapter }{Spun knots and the Satoh tube map}}{37}{section.3.2}\protected@file@percent }
\citation{MR1758871}
\citation{MR2441256}
\@writefile{brf}{\backcite{MR1758871}{{38}{3.2}{section.3.2}}}
\@writefile{brf}{\backcite{MR2441256}{{38}{3.2}{section.3.2}}}
\@writefile{toc}{\contentsline {section}{\numberline {3.3}\textcolor {Chapter }{Cohomology with local coefficients}}{39}{section.3.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {3.4}\textcolor {Chapter }{Distinguishing between two non\texttt  {\symbol  {45}}homeomorphic homotopy equivalent spaces}}{40}{section.3.4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {3.5}\textcolor {Chapter }{ Second homotopy groups of spaces with finite fundamental group}}{40}{section.3.5}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {3.6}\textcolor {Chapter }{Third homotopy groups of simply connected spaces}}{41}{section.3.6}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {3.6.1}\textcolor {Chapter }{First example: Whitehead's certain exact sequence}}{41}{subsection.3.6.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {3.6.2}\textcolor {Chapter }{Second example: the Hopf invariant}}{42}{subsection.3.6.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {3.7}\textcolor {Chapter }{Computing the second homotopy group of a space with infinite fundamental group}}{43}{section.3.7}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {4}\textcolor {Chapter }{Three Manifolds}}{45}{chapter.4}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {4.1}\textcolor {Chapter }{Dehn Surgery}}{45}{section.4.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4.2}\textcolor {Chapter }{Connected Sums}}{46}{section.4.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4.3}\textcolor {Chapter }{Dijkgraaf\texttt  {\symbol  {45}}Witten Invariant}}{46}{section.4.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4.4}\textcolor {Chapter }{Cohomology rings}}{47}{section.4.4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4.5}\textcolor {Chapter }{Linking Form}}{48}{section.4.5}\protected@file@percent }
\citation{reidemeister}
\citation{moise}
\citation{brody}
\citation{przytycki}
\@writefile{toc}{\contentsline {section}{\numberline {4.6}\textcolor {Chapter }{Determining the homeomorphism type of a lens space}}{49}{section.4.6}\protected@file@percent }
\@writefile{brf}{\backcite{reidemeister}{{49}{4.6}{section.4.6}}}
\@writefile{brf}{\backcite{moise}{{49}{4.6}{section.4.6}}}
\@writefile{brf}{\backcite{brody}{{49}{4.6}{section.4.6}}}
\@writefile{brf}{\backcite{przytycki}{{49}{4.6}{section.4.6}}}
\@writefile{toc}{\contentsline {section}{\numberline {4.7}\textcolor {Chapter }{Surgeries on distinct knots can yield homeomorphic manifolds}}{51}{section.4.7}\protected@file@percent }
\citation{lmoser}
\citation{greene}
\@writefile{brf}{\backcite{lmoser}{{52}{4.7}{section.4.7}}}
\@writefile{brf}{\backcite{greene}{{52}{4.7}{section.4.7}}}
\@writefile{toc}{\contentsline {section}{\numberline {4.8}\textcolor {Chapter }{Finite fundamental groups of $3$\texttt  {\symbol  {45}}manifolds}}{52}{section.4.8}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4.9}\textcolor {Chapter }{Poincare's cube manifolds}}{53}{section.4.9}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4.10}\textcolor {Chapter }{There are at least 25 distinct cube manifolds}}{54}{section.4.10}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {4.10.1}\textcolor {Chapter }{Face pairings for 25 distinct cube manifolds}}{56}{subsection.4.10.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {4.10.2}\textcolor {Chapter }{Platonic cube manifolds}}{60}{subsection.4.10.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4.11}\textcolor {Chapter }{There are at most 41 distinct cube manifolds}}{60}{section.4.11}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4.12}\textcolor {Chapter }{There are precisely 18 orientable cube manifolds, of which 9 are spherical and 5 are euclidean}}{62}{section.4.12}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4.13}\textcolor {Chapter }{Cube manifolds with boundary}}{64}{section.4.13}\protected@file@percent }
\citation{thurston}
\@writefile{toc}{\contentsline {section}{\numberline {4.14}\textcolor {Chapter }{Octahedral manifolds}}{65}{section.4.14}\protected@file@percent }
\@writefile{brf}{\backcite{thurston}{{65}{4.14}{section.4.14}}}
\@writefile{toc}{\contentsline {section}{\numberline {4.15}\textcolor {Chapter }{Dodecahedral manifolds}}{65}{section.4.15}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4.16}\textcolor {Chapter }{Prism manifolds}}{66}{section.4.16}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4.17}\textcolor {Chapter }{Bipyramid manifolds}}{67}{section.4.17}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {5}\textcolor {Chapter }{Topological data analysis}}{68}{chapter.5}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {5.1}\textcolor {Chapter }{Persistent homology }}{68}{section.5.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1.1}\textcolor {Chapter }{Background to the data}}{69}{subsection.5.1.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5.2}\textcolor {Chapter }{Mapper clustering}}{69}{section.5.2}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2.1}\textcolor {Chapter }{Background to the data}}{70}{subsection.5.2.1}\protected@file@percent }
\newlabel{pointcloud}{{5.2.1}{70}{\textcolor {Chapter }{Background to the data}}{subsection.5.2.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.3}\textcolor {Chapter }{Some tools for handling pure complexes}}{70}{section.5.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5.4}\textcolor {Chapter }{Digital image analysis and persistent homology}}{71}{section.5.4}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.1}\textcolor {Chapter }{Naive example of image segmentation by automatic thresholding}}{71}{subsection.5.4.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.2}\textcolor {Chapter }{Refining the filtration}}{72}{subsection.5.4.2}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.3}\textcolor {Chapter }{Background to the data}}{72}{subsection.5.4.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5.5}\textcolor {Chapter }{A second example of digital image segmentation}}{72}{section.5.5}\protected@file@percent }
\citation{MartinFTM01}
\@writefile{toc}{\contentsline {section}{\numberline {5.6}\textcolor {Chapter }{A third example of digital image segmentation}}{73}{section.5.6}\protected@file@percent }
\@writefile{brf}{\backcite{MartinFTM01}{{73}{5.6}{section.5.6}}}
\citation{coeurjolly}
\citation{coeurjolly}
\@writefile{toc}{\contentsline {section}{\numberline {5.7}\textcolor {Chapter }{Naive example of digital image contour extraction}}{74}{section.5.7}\protected@file@percent }
\@writefile{brf}{\backcite{coeurjolly}{{74}{5.7}{section.5.7}}}
\@writefile{brf}{\backcite{coeurjolly}{{74}{5.7}{section.5.7}}}
\@writefile{toc}{\contentsline {section}{\numberline {5.8}\textcolor {Chapter }{Alternative approaches to computing persistent homology}}{75}{section.5.8}\protected@file@percent }
\newlabel{secAltPersist}{{5.8}{75}{\textcolor {Chapter }{Alternative approaches to computing persistent homology}}{section.5.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.8.1}\textcolor {Chapter }{Non\texttt  {\symbol  {45}}trivial cup product}}{76}{subsection.5.8.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {5.8.2}\textcolor {Chapter }{Explicit homology generators}}{76}{subsection.5.8.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5.9}\textcolor {Chapter }{Knotted proteins}}{77}{section.5.9}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5.10}\textcolor {Chapter }{Random simplicial complexes}}{78}{section.5.10}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5.11}\textcolor {Chapter }{Computing homology of a clique complex (Vietoris\texttt  {\symbol  {45}}Rips complex) }}{80}{section.5.11}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {6}\textcolor {Chapter }{Group theoretic computations}}{82}{chapter.6}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {6.1}\textcolor {Chapter }{Third homotopy group of a supsension of an Eilenberg\texttt  {\symbol  {45}}MacLane space }}{82}{section.6.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {6.2}\textcolor {Chapter }{Representations of knot quandles}}{82}{section.6.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {6.3}\textcolor {Chapter }{Identifying knots}}{83}{section.6.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {6.4}\textcolor {Chapter }{Aspherical $2$\texttt  {\symbol  {45}}complexes}}{83}{section.6.4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {6.5}\textcolor {Chapter }{Group presentations and homotopical syzygies}}{83}{section.6.5}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {6.6}\textcolor {Chapter }{Bogomolov multiplier}}{85}{section.6.6}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {6.7}\textcolor {Chapter }{Second group cohomology and group extensions}}{85}{section.6.7}\protected@file@percent }
\newlabel{secExtensions}{{6.7}{85}{\textcolor {Chapter }{Second group cohomology and group extensions}}{section.6.7}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6.8}\textcolor {Chapter }{Cocyclic groups: a convenient way of representing certain groups}}{88}{section.6.8}\protected@file@percent }
\newlabel{secCocyclic}{{6.8}{88}{\textcolor {Chapter }{Cocyclic groups: a convenient way of representing certain groups}}{section.6.8}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6.9}\textcolor {Chapter }{Effective group presentations}}{89}{section.6.9}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {6.10}\textcolor {Chapter }{Second group cohomology and cocyclic Hadamard matrices}}{91}{section.6.10}\protected@file@percent }
\newlabel{secHadamard}{{6.10}{91}{\textcolor {Chapter }{Second group cohomology and cocyclic Hadamard matrices}}{section.6.10}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6.11}\textcolor {Chapter }{Third group cohomology and homotopy $2$\texttt  {\symbol  {45}}types}}{91}{section.6.11}\protected@file@percent }
\newlabel{secCat1}{{6.11}{91}{\textcolor {Chapter }{Third group cohomology and homotopy $2$\texttt {\symbol {45}}types}}{section.6.11}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {7}\textcolor {Chapter }{Cohomology of groups (and Lie Algebras)}}{94}{chapter.7}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {7.1}\textcolor {Chapter }{Finite groups }}{94}{section.7.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {7.1.1}\textcolor {Chapter }{Naive homology computation for a very small group}}{94}{subsection.7.1.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {7.1.2}\textcolor {Chapter }{A more efficient homology computation}}{95}{subsection.7.1.2}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {7.1.3}\textcolor {Chapter }{Computation of an induced homology homomorphism}}{95}{subsection.7.1.3}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {7.1.4}\textcolor {Chapter }{Some other finite group homology computations}}{96}{subsection.7.1.4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {7.2}\textcolor {Chapter }{Nilpotent groups}}{97}{section.7.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {7.3}\textcolor {Chapter }{Crystallographic and Almost Crystallographic groups}}{98}{section.7.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {7.4}\textcolor {Chapter }{Arithmetic groups}}{98}{section.7.4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {7.5}\textcolor {Chapter }{Artin groups}}{98}{section.7.5}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {7.6}\textcolor {Chapter }{Graphs of groups}}{99}{section.7.6}\protected@file@percent }
\citation{kuzmin}
\citation{igusa}
\@writefile{toc}{\contentsline {section}{\numberline {7.7}\textcolor {Chapter }{Lie algebra homology and free nilpotent groups}}{100}{section.7.7}\protected@file@percent }
\@writefile{brf}{\backcite{kuzmin}{{100}{7.7}{section.7.7}}}
\@writefile{brf}{\backcite{igusa}{{100}{7.7}{section.7.7}}}
\@writefile{toc}{\contentsline {section}{\numberline {7.8}\textcolor {Chapter }{Cohomology with coefficients in a module}}{101}{section.7.8}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {7.9}\textcolor {Chapter }{Cohomology as a functor of the first variable}}{103}{section.7.9}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {7.10}\textcolor {Chapter }{Cohomology as a functor of the second variable and the long exact coefficient sequence}}{104}{section.7.10}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {7.11}\textcolor {Chapter }{Transfer Homomorphism}}{105}{section.7.11}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {7.12}\textcolor {Chapter }{Cohomology rings of finite fundamental groups of 3\texttt  {\symbol  {45}}manifolds }}{106}{section.7.12}\protected@file@percent }
\newlabel{Secfinitefundman}{{7.12}{106}{\textcolor {Chapter }{Cohomology rings of finite fundamental groups of 3\texttt {\symbol {45}}manifolds }}{section.7.12}{}}
\citation{tomoda}
\@writefile{brf}{\backcite{tomoda}{{107}{7.12}{section.7.12}}}
\citation{dpr}
\citation{horadam}
\@writefile{toc}{\contentsline {section}{\numberline {7.13}\textcolor {Chapter }{Explicit cocycles }}{108}{section.7.13}\protected@file@percent }
\@writefile{brf}{\backcite{dpr}{{108}{7.13}{section.7.13}}}
\@writefile{brf}{\backcite{horadam}{{108}{7.13}{section.7.13}}}
\citation{Webb}
\citation{Webb}
\@writefile{toc}{\contentsline {section}{\numberline {7.14}\textcolor {Chapter }{Quillen's complex and the $p$\texttt  {\symbol  {45}}part of homology }}{111}{section.7.14}\protected@file@percent }
\newlabel{secWebb}{{7.14}{111}{\textcolor {Chapter }{Quillen's complex and the $p$\texttt {\symbol {45}}part of homology }}{section.7.14}{}}
\@writefile{brf}{\backcite{Webb}{{111}{7.14}{section.7.14}}}
\@writefile{brf}{\backcite{Webb}{{111}{7.14}{section.7.14}}}
\@writefile{toc}{\contentsline {section}{\numberline {7.15}\textcolor {Chapter }{Homology of a Lie algebra}}{114}{section.7.15}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {7.16}\textcolor {Chapter }{Covers of Lie algebras}}{114}{section.7.16}\protected@file@percent }
\citation{Wustner}
\@writefile{toc}{\contentsline {subsection}{\numberline {7.16.1}\textcolor {Chapter }{Computing a cover}}{115}{subsection.7.16.1}\protected@file@percent }
\@writefile{brf}{\backcite{Wustner}{{115}{7.16.1}{subsection.7.16.1}}}
\@writefile{toc}{\contentsline {chapter}{\numberline {8}\textcolor {Chapter }{Cohomology rings and Steenrod operations for groups}}{116}{chapter.8}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {8.1}\textcolor {Chapter }{Mod\texttt  {\symbol  {45}}$p$ cohomology rings of finite groups}}{116}{section.8.1}\protected@file@percent }
\citation{Symmonds}
\@writefile{brf}{\backcite{Symmonds}{{117}{8.1}{section.8.1}}}
\@writefile{toc}{\contentsline {subsection}{\numberline {8.1.1}\textcolor {Chapter }{Ring presentations (for the commutative $p=2$ case)}}{117}{subsection.8.1.1}\protected@file@percent }
\citation{Symmonds}
\@writefile{toc}{\contentsline {section}{\numberline {8.2}\textcolor {Chapter }{Poincare Series for Mod\texttt  {\symbol  {45}}$p$ cohomology}}{118}{section.8.2}\protected@file@percent }
\@writefile{brf}{\backcite{Symmonds}{{118}{8.2}{section.8.2}}}
\@writefile{toc}{\contentsline {section}{\numberline {8.3}\textcolor {Chapter }{Functorial ring homomorphisms in Mod\texttt  {\symbol  {45}}$p$ cohomology}}{119}{section.8.3}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {8.3.1}\textcolor {Chapter }{Testing homomorphism properties}}{120}{subsection.8.3.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {8.3.2}\textcolor {Chapter }{Testing functorial properties}}{120}{subsection.8.3.2}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {8.3.3}\textcolor {Chapter }{Computing with larger groups}}{121}{subsection.8.3.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {8.4}\textcolor {Chapter }{Steenrod operations for finite $2$\texttt  {\symbol  {45}}groups}}{122}{section.8.4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {8.5}\textcolor {Chapter }{Steenrod operations on the classifying space of a finite $p$\texttt  {\symbol  {45}}group}}{123}{section.8.5}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {8.6}\textcolor {Chapter }{Mod\texttt  {\symbol  {45}}$p$ cohomology rings of crystallographic groups}}{123}{section.8.6}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {8.6.1}\textcolor {Chapter }{Poincare series for crystallographic groups}}{123}{subsection.8.6.1}\protected@file@percent }
\citation{liuye}
\citation{liuyegithub}
\citation{liuye}
\citation{liuyegithub}
\@writefile{toc}{\contentsline {subsection}{\numberline {8.6.2}\textcolor {Chapter }{Mod $2$ cohomology rings of $3$\texttt  {\symbol  {45}}dimensional crystallographic groups}}{125}{subsection.8.6.2}\protected@file@percent }
\@writefile{brf}{\backcite{liuye}{{125}{8.6.2}{subsection.8.6.2}}}
\@writefile{brf}{\backcite{liuyegithub}{{125}{8.6.2}{subsection.8.6.2}}}
\@writefile{brf}{\backcite{liuye}{{125}{8.6.2}{subsection.8.6.2}}}
\@writefile{brf}{\backcite{liuyegithub}{{125}{8.6.2}{subsection.8.6.2}}}
\citation{liuye}
\@writefile{brf}{\backcite{liuye}{{126}{8.6.2}{subsection.8.6.2}}}
\@writefile{toc}{\contentsline {chapter}{\numberline {9}\textcolor {Chapter }{Bredon homology}}{127}{chapter.9}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {9.1}\textcolor {Chapter }{Davis complex}}{127}{section.9.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {9.2}\textcolor {Chapter }{Arithmetic groups}}{127}{section.9.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {9.3}\textcolor {Chapter }{Crystallographic groups}}{128}{section.9.3}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {10}\textcolor {Chapter }{Chain Complexes}}{129}{chapter.10}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {10.1}\textcolor {Chapter }{Chain complex of a simplicial complex and simplicial pair}}{129}{section.10.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {10.2}\textcolor {Chapter }{Chain complex of a cubical complex and cubical pair}}{130}{section.10.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {10.3}\textcolor {Chapter }{Chain complex of a regular CW\texttt  {\symbol  {45}}complex}}{131}{section.10.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {10.4}\textcolor {Chapter }{Chain Maps of simplicial and regular CW maps}}{132}{section.10.4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {10.5}\textcolor {Chapter }{Constructions for chain complexes}}{132}{section.10.5}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {10.6}\textcolor {Chapter }{Filtered chain complexes}}{133}{section.10.6}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {10.7}\textcolor {Chapter }{Sparse chain complexes}}{134}{section.10.7}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {11}\textcolor {Chapter }{Resolutions}}{136}{chapter.11}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{resolutions}{{11}{136}{\textcolor {Chapter }{Resolutions}}{chapter.11}{}}
\@writefile{toc}{\contentsline {section}{\numberline {11.1}\textcolor {Chapter }{Resolutions for small finite groups}}{136}{section.11.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.2}\textcolor {Chapter }{Resolutions for very small finite groups}}{136}{section.11.2}\protected@file@percent }
\citation{swan2}
\citation{kholodna}
\citation{johnson}
\@writefile{brf}{\backcite{swan2}{{137}{11.2}{section.11.2}}}
\@writefile{brf}{\backcite{kholodna}{{137}{11.2}{section.11.2}}}
\@writefile{brf}{\backcite{johnson}{{137}{11.2}{section.11.2}}}
\@writefile{toc}{\contentsline {section}{\numberline {11.3}\textcolor {Chapter }{Resolutions for finite groups acting on orbit polytopes}}{138}{section.11.3}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.4}\textcolor {Chapter }{Minimal resolutions for finite $p$\texttt  {\symbol  {45}}groups over $\mathbb  F_p$}}{139}{section.11.4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.5}\textcolor {Chapter }{Resolutions for abelian groups}}{139}{section.11.5}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.6}\textcolor {Chapter }{Resolutions for nilpotent groups}}{140}{section.11.6}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.7}\textcolor {Chapter }{Resolutions for groups with subnormal series}}{141}{section.11.7}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.8}\textcolor {Chapter }{Resolutions for groups with normal series}}{141}{section.11.8}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.9}\textcolor {Chapter }{Resolutions for polycyclic (almost) crystallographic groups }}{141}{section.11.9}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.10}\textcolor {Chapter }{Resolutions for Bieberbach groups }}{142}{section.11.10}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.11}\textcolor {Chapter }{Resolutions for arbitrary crystallographic groups}}{143}{section.11.11}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.12}\textcolor {Chapter }{Resolutions for crystallographic groups admitting cubical fundamental domain}}{143}{section.11.12}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.13}\textcolor {Chapter }{Resolutions for Coxeter groups }}{144}{section.11.13}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.14}\textcolor {Chapter }{Resolutions for Artin groups }}{144}{section.11.14}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.15}\textcolor {Chapter }{Resolutions for $G=SL_2(\mathbb  Z[1/m])$}}{145}{section.11.15}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.16}\textcolor {Chapter }{Resolutions for selected groups $G=SL_2( {\mathcal  O}(\mathbb  Q(\sqrt  {d}) )$}}{145}{section.11.16}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.17}\textcolor {Chapter }{Resolutions for selected groups $G=PSL_2( {\mathcal  O}(\mathbb  Q(\sqrt  {d}) )$}}{145}{section.11.17}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.18}\textcolor {Chapter }{Resolutions for a few higher\texttt  {\symbol  {45}}dimensional arithmetic groups }}{146}{section.11.18}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.19}\textcolor {Chapter }{Resolutions for finite\texttt  {\symbol  {45}}index subgroups }}{146}{section.11.19}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.20}\textcolor {Chapter }{Simplifying resolutions }}{147}{section.11.20}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.21}\textcolor {Chapter }{Resolutions for graphs of groups and for groups with aspherical presentations }}{147}{section.11.21}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {11.22}\textcolor {Chapter }{Resolutions for $\mathbb  FG$\texttt  {\symbol  {45}}modules }}{148}{section.11.22}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {12}\textcolor {Chapter }{Simplicial groups}}{149}{chapter.12}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{chapSimplicialGroups}{{12}{149}{\textcolor {Chapter }{Simplicial groups}}{chapter.12}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12.1}\textcolor {Chapter }{Crossed modules}}{149}{section.12.1}\protected@file@percent }
\newlabel{secCrossedModules}{{12.1}{149}{\textcolor {Chapter }{Crossed modules}}{section.12.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12.2}\textcolor {Chapter }{Eilenberg\texttt  {\symbol  {45}}MacLane spaces as simplicial groups (not recommended)}}{150}{section.12.2}\protected@file@percent }
\newlabel{eilennot}{{12.2}{150}{\textcolor {Chapter }{Eilenberg\texttt {\symbol {45}}MacLane spaces as simplicial groups (not recommended)}}{section.12.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12.3}\textcolor {Chapter }{Eilenberg\texttt  {\symbol  {45}}MacLane spaces as simplicial free abelian groups (recommended)}}{150}{section.12.3}\protected@file@percent }
\newlabel{eilen}{{12.3}{150}{\textcolor {Chapter }{Eilenberg\texttt {\symbol {45}}MacLane spaces as simplicial free abelian groups (recommended)}}{section.12.3}{}}
\citation{hatcher}
\@writefile{toc}{\contentsline {section}{\numberline {12.4}\textcolor {Chapter }{Elementary theoretical information on $H^\ast (K(\pi ,n),\mathbb  Z)$}}{152}{section.12.4}\protected@file@percent }
\@writefile{brf}{\backcite{hatcher}{{153}{12.4}{section.12.4}}}
\@writefile{toc}{\contentsline {section}{\numberline {12.5}\textcolor {Chapter }{The first three non\texttt  {\symbol  {45}}trivial homotopy groups of spheres}}{153}{section.12.5}\protected@file@percent }
\newlabel{firstthree}{{12.5}{153}{\textcolor {Chapter }{The first three non\texttt {\symbol {45}}trivial homotopy groups of spheres}}{section.12.5}{}}
\citation{brownloday}
\@writefile{toc}{\contentsline {section}{\numberline {12.6}\textcolor {Chapter }{The first two non\texttt  {\symbol  {45}}trivial homotopy groups of the suspension and double suspension of a $K(G,1)$}}{154}{section.12.6}\protected@file@percent }
\newlabel{firsttwo}{{12.6}{154}{\textcolor {Chapter }{The first two non\texttt {\symbol {45}}trivial homotopy groups of the suspension and double suspension of a $K(G,1)$}}{section.12.6}{}}
\@writefile{brf}{\backcite{brownloday}{{154}{12.6}{section.12.6}}}
\@writefile{toc}{\contentsline {section}{\numberline {12.7}\textcolor {Chapter }{Postnikov towers and $\pi _5(S^3)$ }}{154}{section.12.7}\protected@file@percent }
\newlabel{postnikov2}{{12.7}{154}{\textcolor {Chapter }{Postnikov towers and $\pi _5(S^3)$ }}{section.12.7}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12.8}\textcolor {Chapter }{Towards $\pi _4(\Sigma K(G,1))$ }}{156}{section.12.8}\protected@file@percent }
\newlabel{postnikov}{{12.8}{156}{\textcolor {Chapter }{Towards $\pi _4(\Sigma K(G,1))$ }}{section.12.8}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12.9}\textcolor {Chapter }{Enumerating homotopy 2\texttt  {\symbol  {45}}types}}{157}{section.12.9}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {12.10}\textcolor {Chapter }{Identifying cat$^1$\texttt  {\symbol  {45}}groups of low order}}{158}{section.12.10}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {12.11}\textcolor {Chapter }{Identifying crossed modules of low order}}{159}{section.12.11}\protected@file@percent }
\citation{eichler}
\citation{shimura}
\citation{stein}
\@writefile{toc}{\contentsline {chapter}{\numberline {13}\textcolor {Chapter }{Congruence Subgroups, Cuspidal Cohomology and Hecke Operators}}{161}{chapter.13}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {13.1}\textcolor {Chapter }{Eichler\texttt  {\symbol  {45}}Shimura isomorphism}}{161}{section.13.1}\protected@file@percent }
\newlabel{sec:EichlerShimura}{{13.1}{161}{\textcolor {Chapter }{Eichler\texttt {\symbol {45}}Shimura isomorphism}}{section.13.1}{}}
\@writefile{brf}{\backcite{eichler}{{161}{13.1}{section.13.1}}}
\@writefile{brf}{\backcite{shimura}{{161}{13.1}{section.13.1}}}
\citation{wieser}
\@writefile{brf}{\backcite{stein}{{162}{13.1}{section.13.1}}}
\@writefile{brf}{\backcite{wieser}{{162}{13.1}{section.13.1}}}
\@writefile{toc}{\contentsline {section}{\numberline {13.2}\textcolor {Chapter }{Generators for $SL_2(\mathbb  Z)$ and the cubic tree}}{162}{section.13.2}\protected@file@percent }
\citation{ellisharrisskoldberg}
\@writefile{toc}{\contentsline {section}{\numberline {13.3}\textcolor {Chapter }{One\texttt  {\symbol  {45}}dimensional fundamental domains and generators for congruence subgroups}}{163}{section.13.3}\protected@file@percent }
\@writefile{brf}{\backcite{ellisharrisskoldberg}{{163}{13.3}{section.13.3}}}
\citation{kulkarni}
\citation{buiellis}
\@writefile{brf}{\backcite{kulkarni}{{164}{13.3}{section.13.3}}}
\@writefile{toc}{\contentsline {section}{\numberline {13.4}\textcolor {Chapter }{Cohomology of congruence subgroups}}{164}{section.13.4}\protected@file@percent }
\@writefile{brf}{\backcite{buiellis}{{164}{13.4}{section.13.4}}}
\@writefile{toc}{\contentsline {subsection}{\numberline {13.4.1}\textcolor {Chapter }{Cohomology with rational coefficients}}{166}{subsection.13.4.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {13.5}\textcolor {Chapter }{Cuspidal cohomology}}{166}{section.13.5}\protected@file@percent }
\citation{stein}
\@writefile{toc}{\contentsline {section}{\numberline {13.6}\textcolor {Chapter }{Hecke operators on forms of weight 2}}{168}{section.13.6}\protected@file@percent }
\@writefile{brf}{\backcite{stein}{{168}{13.6}{section.13.6}}}
\citation{stein}
\citation{atkinlehner}
\@writefile{toc}{\contentsline {section}{\numberline {13.7}\textcolor {Chapter }{Hecke operators on forms of weight $ \ge 2$}}{169}{section.13.7}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {13.8}\textcolor {Chapter }{Reconstructing modular forms from cohomology computations}}{169}{section.13.8}\protected@file@percent }
\@writefile{brf}{\backcite{stein}{{169}{13.8}{section.13.8}}}
\@writefile{brf}{\backcite{atkinlehner}{{169}{13.8}{section.13.8}}}
\@writefile{toc}{\contentsline {section}{\numberline {13.9}\textcolor {Chapter }{The Picard group}}{171}{section.13.9}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {13.10}\textcolor {Chapter }{Bianchi groups}}{172}{section.13.10}\protected@file@percent }
\citation{swan}
\citation{rahmthesis}
\citation{schoennenbeck}
\@writefile{brf}{\backcite{swan}{{173}{13.10}{section.13.10}}}
\@writefile{brf}{\backcite{rahmthesis}{{173}{13.10}{section.13.10}}}
\@writefile{brf}{\backcite{schoennenbeck}{{173}{13.10}{section.13.10}}}
\citation{sengun}
\citation{Schwermer}
\citation{Vogtmann}
\citation{Berkove00}
\citation{Berkove06}
\citation{Rahm11}
\citation{Rahm13}
\citation{Rahm13a}
\citation{Rahm20}
\citation{Rahm11}
\@writefile{brf}{\backcite{sengun}{{174}{13.10}{section.13.10}}}
\@writefile{toc}{\contentsline {section}{\numberline {13.11}\textcolor {Chapter }{(Co)homology of Bianchi groups and $SL_2({\cal  O}_{-d})$}}{174}{section.13.11}\protected@file@percent }
\@writefile{brf}{\backcite{Schwermer}{{174}{13.11}{section.13.11}}}
\@writefile{brf}{\backcite{Vogtmann}{{174}{13.11}{section.13.11}}}
\@writefile{brf}{\backcite{Berkove00}{{174}{13.11}{section.13.11}}}
\@writefile{brf}{\backcite{Berkove06}{{174}{13.11}{section.13.11}}}
\@writefile{brf}{\backcite{Rahm11}{{174}{13.11}{section.13.11}}}
\@writefile{brf}{\backcite{Rahm13}{{174}{13.11}{section.13.11}}}
\@writefile{brf}{\backcite{Rahm13a}{{174}{13.11}{section.13.11}}}
\@writefile{brf}{\backcite{Rahm20}{{174}{13.11}{section.13.11}}}
\@writefile{brf}{\backcite{Rahm11}{{174}{13.11}{section.13.11}}}
\citation{Schwermer}
\@writefile{brf}{\backcite{Schwermer}{{175}{13.11}{section.13.11}}}
\citation{Rahm11}
\citation{Rahm11}
\@writefile{brf}{\backcite{Rahm11}{{176}{13.11}{section.13.11}}}
\@writefile{brf}{\backcite{Rahm11}{{176}{13.11}{section.13.11}}}
\@writefile{toc}{\contentsline {section}{\numberline {13.12}\textcolor {Chapter }{Some other infinite matrix groups}}{179}{section.13.12}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {13.13}\textcolor {Chapter }{Ideals and finite quotient groups}}{181}{section.13.13}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {13.14}\textcolor {Chapter }{Congruence subgroups for ideals}}{182}{section.13.14}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {13.15}\textcolor {Chapter }{First homology}}{183}{section.13.15}\protected@file@percent }
\citation{bergeron}
\@writefile{brf}{\backcite{bergeron}{{184}{13.15}{section.13.15}}}
\citation{swanB}
\@writefile{toc}{\contentsline {chapter}{\numberline {14}\textcolor {Chapter }{Fundamental domains for Bianchi groups}}{186}{chapter.14}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {14.1}\textcolor {Chapter }{Bianchi groups}}{186}{section.14.1}\protected@file@percent }
\@writefile{brf}{\backcite{swanB}{{186}{14.1}{section.14.1}}}
\@writefile{toc}{\contentsline {section}{\numberline {14.2}\textcolor {Chapter }{Swan's description of a fundamental domain}}{186}{section.14.2}\protected@file@percent }
\citation{swanB}
\citation{swanB}
\@writefile{brf}{\backcite{swanB}{{187}{14.2}{section.14.2}}}
\@writefile{toc}{\contentsline {section}{\numberline {14.3}\textcolor {Chapter }{Computing a fundamental domain}}{187}{section.14.3}\protected@file@percent }
\@writefile{brf}{\backcite{swanB}{{187}{14.3}{section.14.3}}}
\@writefile{toc}{\contentsline {section}{\numberline {14.4}\textcolor {Chapter }{Examples}}{187}{section.14.4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {14.5}\textcolor {Chapter }{Establishing correctness of a fundamental domain}}{188}{section.14.5}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {14.6}\textcolor {Chapter }{Computing a free resolution for $SL_2({\mathcal  O}_{-d})$}}{189}{section.14.6}\protected@file@percent }
\citation{Gangl}
\@writefile{toc}{\contentsline {section}{\numberline {14.7}\textcolor {Chapter }{Some sanity checks}}{190}{section.14.7}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {14.7.1}\textcolor {Chapter }{Equivariant Euler characteristic}}{190}{subsection.14.7.1}\protected@file@percent }
\@writefile{brf}{\backcite{Gangl}{{190}{14.7.1}{subsection.14.7.1}}}
\@writefile{toc}{\contentsline {subsection}{\numberline {14.7.2}\textcolor {Chapter }{Boundary squares to zero}}{191}{subsection.14.7.2}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {14.7.3}\textcolor {Chapter }{Compare different algorithms or implementations}}{191}{subsection.14.7.3}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {14.7.4}\textcolor {Chapter }{Compare geometry to algebra}}{192}{subsection.14.7.4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {14.8}\textcolor {Chapter }{Group presentations}}{192}{section.14.8}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {14.9}\textcolor {Chapter }{Finite index subgroups}}{193}{section.14.9}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {15}\textcolor {Chapter }{Parallel computation}}{195}{chapter.15}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {15.1}\textcolor {Chapter }{An embarassingly parallel computation}}{195}{section.15.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {15.2}\textcolor {Chapter }{A non\texttt  {\symbol  {45}}embarassingly parallel computation}}{195}{section.15.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {15.3}\textcolor {Chapter }{Parallel persistent homology}}{197}{section.15.3}\protected@file@percent }
\@writefile{toc}{\contentsline {chapter}{\numberline {16}\textcolor {Chapter }{Regular CW\texttt  {\symbol  {45}}structure on knots (written by Kelvin Killeen)}}{198}{chapter.16}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {16.1}\textcolor {Chapter }{Knot complements in the 3\texttt  {\symbol  {45}}ball}}{198}{section.16.1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {16.2}\textcolor {Chapter }{Tubular neighbourhoods}}{199}{section.16.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {16.3}\textcolor {Chapter }{Knotted surface complements in the 4\texttt  {\symbol  {45}}ball}}{202}{section.16.3}\protected@file@percent }
\bibstyle{alpha}
\bibdata{mybib.xml}
\bibcite{atkinlehner}{AL70}
\bibcite{schoennenbeck}{BCNS15}
\bibcite{buiellis}{BE14}
\bibcite{Berkove00}{Ber00}
\bibcite{Berkove06}{Ber06}
\bibcite{bergeron}{Ber16}
\bibcite{brownloday}{BL87}
\bibcite{Rahm20}{BLR20}
\bibcite{brody}{Bro60}
\bibcite{coeurjolly}{CKL14}
\bibcite{dpr}{DPR91}
\bibcite{Gangl}{DSGG{$^{+}$}16}
\bibcite{ellisharrisskoldberg}{EHS06}
\bibcite{eichler}{Eic57}
\bibcite{goncalves}{GM15}
\bibcite{greene}{Gre13}
\bibcite{hatcher}{Hat01}
\bibcite{horadam}{Hor00}
\bibcite{igusa}{IO01}
\bibcite{johnson}{Joh16}
\bibcite{MR2441256}{KFM08}
\bibcite{kholodna}{Kho01}
\bibcite{kuzmin}{KS98}
\bibcite{ksontini}{Kso00}
\bibcite{kulkarni}{Kul91}
\bibcite{liuye}{LY24a}
\bibcite{liuyegithub}{LY24b}
\bibcite{MartinFTM01}{MFTM01}
\bibcite{milnor}{Mil58}
\bibcite{moise}{Moi52}
\bibcite{lmoser}{Mos71}
\bibcite{przytycki}{PY03}
\bibcite{rahmthesis}{Rah10}
\bibcite{Rahm13a}{Rah13a}
\bibcite{Rahm13}{Rah13b}
\bibcite{reidemeister}{Rei35}
\bibcite{Rahm11}{RF13}
\bibcite{MR1758871}{Sat00}
\bibcite{sengun}{Sen11}
\bibcite{shimura}{Shi59}
\bibcite{spreerkhuenel}{SK11}
\bibcite{stein}{Ste07}
\bibcite{Schwermer}{SV83}
\bibcite{swan2}{Swa60}
\bibcite{swan}{Swa71a}
\bibcite{swanB}{Swa71b}
\bibcite{Symmonds}{Sym10}
\bibcite{thurston}{Thu02}
\bibcite{tomoda}{TZ08}
\bibcite{Vogtmann}{Vog85}
\bibcite{Webb}{Web87}
\bibcite{wieser}{Wie78}
\bibcite{Wustner}{Wue92}
\@writefile{toc}{\contentsline {chapter}{References}{212}{chapter*.2}\protected@file@percent }